UCSB COMPUTER SCIENCE DEPARTMENT/CSE IGERT PRESENTS:

Monday, January 30, 2012

3:00 – 4:00 PM

Computer Science Conference Room, Harold Frank Hall Rm. 1132

HOST: Linda Petzold

SPEAKER: Gilbert Strang

MIT

Title: Balanced Splitting Methods / Infinite Matrices

Abstract:

Most of this talk is about a problem that constantly arises in

scientific computing.

The last part is about an algebra problem — for finite and infinite

matrices.

Differential equations often have diffusion and advection and reaction

terms. Those are

treated differently, and separately when possible. Diffusion might be

implicit and advection

explicit. Reaction is highly nonlinear, but local. If we separate the

transport terms T(u)

from the reaction terms R(u), we may “split” each time step into

separate integrations.

Since they don’t commute, the overall step has only first-order

accuracy. But second order

is achieved by a half-step based on T, a full step based on R, and a

half-step based on T.

Problem: An error can appear in the steady state. The solution to T(u)+

R(u) = 0 may not

solve T(u) = 0 and R(u) = 0 separately. Solution: Adjust to T(u) + c_n

and R(u) – c_n by a

balancing vector c_n at each step. If we choose c_n = (R(u_n) -

T(u_n))/2 then each part can

converge to the correct steady state. But stability becomes harder to

ensure — and Ray Speth

has created a “rebalanced splitting” that is equally accurate and much

more stable.

For two problems on banded doubly infinite matrices there is progress to

report:

1. Is the triangular factorization A = LPU still possible ? Notice the

position of P !!

2. Which is the ‘main diagonal’ of that infinite matrix ?

Bio:

Gilbert Strang was an undergraduate at MIT and a Rhodes Scholar at

Balliol College, Oxford. His Ph.D. was from UCLA and since then he has

taught at MIT. He is a Fellow of the American Academy of Arts and

Sciences and the National Academy of Sciences.. Professor Strang has

published eight textbooks including Introduction to Linear Algebra and

Computational Science and Engineering.

He was the President of SIAM during 1999 and 2000, and Chair of the

Joint Policy Board for Mathematics. He received the von Neumann Medal

for computational mechanics, and the Henrici Prize for applied analysis.

The first Su Buchin Prize and the Haimo Prize from the MAA were awarded

for his contributions to teaching around the world.

His video lectures on linear algebra and CSE are on ocw.mit.edu